Problem: What is the inverse of the function $g(x)=-\dfrac{2}{3}x-5$ ? $g^{-1}(x)=$
Explanation: Let's start by replacing $g(x)$ with $y$. $y=-\dfrac{2}{3}x-5$ If a function contains the point $(a,b)$, the inverse of that function contains the point $(b,a)$. So if we swap the position of $x$ and $y$ in the equation, we get the inverse relationship. In this case, the function is $y=-\dfrac{2}{3}x-5$, so the inverse relationship is $x=-\dfrac{2}{3}y-5$. Solving this equation for $y$ will give us an expression for $g^{-1}(x)$. $\begin{aligned} x&=-\dfrac{2}{3}y-5\\\\ x+5&=-\dfrac{2}{3}y\\\\ -\dfrac{3}{2}(x+5)&=y\\\\\\ \end{aligned}$ The inverse of the function is $g^{-1}(x)=-\dfrac{3}{2}(x+5)$. [I saw someone solve this problem by originally solving for x. Were they wrong?]